1. Peter Millington
School of Physics and Astronomy
University of Manchester
A Casual Acausality in Feynman Diagrams in collaboration with Bob Dickinson and Jeff Forshaw Consortium Meeting 18 Sep 2013 University of Manchester

2. Causality in Quantum Field Theory
Quantum field theory is causal …
… because we said so.
At the level of field operators, microcausality is imposed via the canonical commutation relations:
𝜑 𝑥 , 𝜑 𝑦 =0, (𝑥−𝑦) 2 <0

3. What about the path integral?
Where does this causal structure go when we move from operators to eigenvalues?
Markovian property of the path integral kernel.
The Feynman prescription and the Wick rotation to 𝑅 4 are acausal? 𝑖 ∆ 1 𝑥,𝑦 = 𝑚 4𝜋 𝑥−𝑦 2 𝜃 ( 𝑥−𝑦) 2 𝑌 1 (𝑚 𝑥−𝑦 2 )
+ 𝑚 4𝜋 − 𝑥−𝑦 2 𝜃 − 𝑥−𝑦 2 𝐾 1 (𝑚 − 𝑥−𝑦 2 ) [see eg. Greiner and Reinhard, Field Quantization]
Propagators are not Markovian? ( 𝜕 𝑧 0 = 𝜕 𝑧 0 − 𝜕 𝑧 0 ) 𝑖 ∆ F 𝑥,𝑦 =𝑖 sgn( 𝑥 0 − 𝑦 0 ) d 3 𝒛 𝑖 ∆ 𝐹 (𝑥,𝑧) 𝜕 𝑧 0 𝑖 ∆ 𝐹 (𝑧,𝑦)

4. A problem with S-matrix theory?𝑐𝑡 𝑥 𝑖𝑛
𝑜𝑢𝑡 +∞ −∞
No.
The adiabatic hypothesis:
in and out asymptotic states are defined in the infinitely-distant past and future all regions of spacetime are causally connected.

5. So what is the problem?
How are we to calculate scattering amplitudes over finite domains in spacetime?
Finite hypervolumes time- and length-scales energy and momentum scales characteristic temperature
Physical sources the language of density operators
In thermal field theory, physical reaction rates are obtained from the absorptive parts of retarded functions. [Kobes, Phys. Rev. D , van Eijck and van Weert, Phys. Lett. B278 (1992)]

6. Something leading …
Introduce a physical source, localized at 𝑧 𝜇 and normalized to emit/absorb a single particle [see e.g. Peskin and Schroeder, QFT]: 𝐾 𝑥 =𝐴 𝛿 4 (𝑥−𝑧), 𝐴= [𝑖 ∆ F 0 ] −1/2
Solution to non-interacting Klein-Gordon equation, assuming Feynman pole prescription: 𝜕 𝜇 𝜕 𝜇 + 𝑚 2 𝜑 𝑥 =𝐾 𝑥 , 𝜑 𝑥 = 𝜑 0 𝑥 −𝐴 ∆ F 𝑥,𝑧
Time-ordered product of fields: T[𝜑(𝑥)𝜑(𝑦)] =𝑖 ∆ F 𝑥,𝑦 − 𝐴 2 𝑖 ∆ F 𝑥,𝑧 𝑖 ∆ F (𝑧,𝑦)

7. In summary … Quantum field theory is causal.
Scattering matrix theory is causal, albeit in a “by default” kind of way.
Want manifestly causal scattering amplitudes over finite domains in spacetime.
The Feynman propagator is not the object that we should use to connect to physical sources …